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Properties

Label 1-184-184.101-r0-0-0
Degree $1$
Conductor $184$
Sign $-0.303 - 0.952i$
Analytic cond. $0.854492$
Root an. cond. $0.854492$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.142 − 0.989i)3-s + (0.959 − 0.281i)5-s + (−0.654 − 0.755i)7-s + (−0.959 − 0.281i)9-s + (−0.841 − 0.540i)11-s + (0.654 − 0.755i)13-s + (−0.142 − 0.989i)15-s + (0.415 + 0.909i)17-s + (−0.415 + 0.909i)19-s + (−0.841 + 0.540i)21-s + (0.841 − 0.540i)25-s + (−0.415 + 0.909i)27-s + (−0.415 − 0.909i)29-s + (−0.142 − 0.989i)31-s + (−0.654 + 0.755i)33-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)3-s + (0.959 − 0.281i)5-s + (−0.654 − 0.755i)7-s + (−0.959 − 0.281i)9-s + (−0.841 − 0.540i)11-s + (0.654 − 0.755i)13-s + (−0.142 − 0.989i)15-s + (0.415 + 0.909i)17-s + (−0.415 + 0.909i)19-s + (−0.841 + 0.540i)21-s + (0.841 − 0.540i)25-s + (−0.415 + 0.909i)27-s + (−0.415 − 0.909i)29-s + (−0.142 − 0.989i)31-s + (−0.654 + 0.755i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $-0.303 - 0.952i$
Analytic conductor: \(0.854492\)
Root analytic conductor: \(0.854492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{184} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 184,\ (0:\ ),\ -0.303 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6968891996 - 0.9536704216i\)
\(L(\frac12)\) \(\approx\) \(0.6968891996 - 0.9536704216i\)
\(L(1)\) \(\approx\) \(0.9620953385 - 0.5633254435i\)
\(L(1)\) \(\approx\) \(0.9620953385 - 0.5633254435i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.142 - 0.989i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (-0.654 - 0.755i)T \)
11 \( 1 + (-0.841 - 0.540i)T \)
13 \( 1 + (0.654 - 0.755i)T \)
17 \( 1 + (0.415 + 0.909i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (-0.415 - 0.909i)T \)
31 \( 1 + (-0.142 - 0.989i)T \)
37 \( 1 + (0.959 + 0.281i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (0.654 - 0.755i)T \)
61 \( 1 + (0.142 + 0.989i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.415 - 0.909i)T \)
79 \( 1 + (-0.654 + 0.755i)T \)
83 \( 1 + (0.959 + 0.281i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (-0.959 + 0.281i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.57463423513669554949645758379, −26.284753392618237240682991669453, −25.76462847781086922147543858367, −25.10265820357303997562509807997, −23.49991950186495309012361645615, −22.512444112512101698354953418615, −21.64105967727270817846103359004, −21.08397322513194063244338043308, −20.02071578718795468900405490999, −18.68770249118932871941738515606, −17.89275434319889985137669435095, −16.59109000967090405422003163243, −15.83151260454396658205891026154, −14.85914549967518599616764601486, −13.83723712463730760471591228610, −12.81679043729493410360038264570, −11.37284489752967080046491878029, −10.32285523969659807991549929815, −9.4443616267993082441623860083, −8.74511145139117907922746099414, −6.92988257599812790496245593077, −5.72340848411304159833351504849, −4.83485619680108484189756254765, −3.20205107337060091642262763985, −2.2676367584046806226324581924, 0.94334155593778574420234908139, 2.34554811892208599498842581444, 3.65939484758039559104757333642, 5.69652897425600959681429810495, 6.211472209748025078776314657651, 7.654793908645833797235632227470, 8.5259428371758233994599148834, 9.94527732844317663986564559674, 10.817910513933700395293932340112, 12.43150491578497154809873995110, 13.27910946484141931636682549401, 13.666279905915596658299431470500, 15.02304476420089655459737011530, 16.57266687177623017769642997319, 17.20402311298539330505766953916, 18.342513087409489916614896559411, 19.0438112976914205547221016063, 20.29214097673253909081100854003, 20.9482325574793432768864012248, 22.30611768264087628064400491584, 23.35340105071439388831896518411, 24.01829569624949357574609630004, 25.2600575208361041548947649149, 25.69661095185958233082188769145, 26.67680908217326246289956334080

Graph of the $Z$-function along the critical line